The moment is here: a generalized class of estimators for fuzzy regression discontinuity designs

Stuart Lane

arxiv: 2511.03424 · v2 · submitted 2025-11-05 · 💰 econ.EM

The moment is here: a generalized class of estimators for fuzzy regression discontinuity designs

Stuart Lane This is my paper

Pith reviewed 2026-05-18 01:43 UTC · model grok-4.3

classification 💰 econ.EM

keywords fuzzy regression discontinuitylocal polynomialsfinite momentstuning parametereconometric estimationtreatment effects

0 comments

The pith

The standard fuzzy regression discontinuity estimator has no finite moments, but a generalized version with a tuning parameter does.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the conventional fuzzy regression discontinuity estimator, formed as a ratio of local polynomial regressions, lacks any finite integer moments no matter the polynomial order, kernel, or bandwidth chosen. This leads to heavy tails and unreliable behavior especially in small samples or when the jump in treatment probability is small. To address this, the author introduces a broader family of estimators controlled by one tuning parameter that keeps all finite moments from the underlying data intact. This family includes the usual fuzzy estimator and the sharp regression discontinuity estimator as special cases. Simulations indicate that fixed choices for the tuning parameter can reduce bias and error measures while maintaining good confidence interval coverage, as shown in an application to class size effects.

Core claim

The standard fuzzy regression discontinuity estimator does not possess any finite integer moments, regardless of local polynomial degree, kernel function, or bandwidth. A generalized class of estimators indexed by a single tuning parameter preserves all finite moments from the data and nests both the standard FRD and sharp SRD estimators.

What carries the argument

A generalized class of fuzzy regression discontinuity estimators defined through a single tuning parameter that modifies the ratio structure to ensure finite moments.

If this is right

Simple fixed values of the tuning parameter reduce median bias, median absolute deviation, and root mean squared error compared to the standard estimator.
Confidence intervals based on the new estimators typically achieve reliable coverage in small samples.
The generalized estimators can be applied to real data such as class size effects on educational attainment with improved stability.
The approach maintains the consistency properties of local polynomial estimators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Researchers working with limited data or weak discontinuities in treatment probability could adopt these estimators to avoid issues with undefined moments.
Similar moment-preserving adjustments might apply to other ratio estimators common in econometrics and statistics.
Optimal selection of the tuning parameter could be explored further to balance bias and variance in specific applications.

Load-bearing premise

That the tuning parameter can be set in a way that guarantees finite moments from the data without violating the local polynomial consistency of the estimator.

What would settle it

A direct calculation or simulation showing that the standard FRD estimator has a finite first moment for some choice of kernel and bandwidth would falsify the claim that it has none.

Figures

Figures reproduced from arXiv: 2511.03424 by Stuart Lane.

**Figure 4.1.** Figure 4.1: Sampling distributions of estimators (a) n = 300 Estimated treatment effect -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 D e n sity 0 0.5 1 1.5 2 2.5 3 3.5 6 = $(4) 6 = $(1) 6 = 1 Cauchy Normal (b) n = 600 Estimated treatment effect -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 D e n sity 0 0.5 1 1.5 2 2.5 3 3.5 6 = $(4) 6 = $(1) 6 = 1 Cauchy Normal Sampling distributions for ˆτΛ(4),1 , ˆτΛ(1),1 and ˆτ1,1. Th… view at source ↗

read the original abstract

The standard fuzzy regression discontinuity (FRD) estimator is a ratio of differences of local polynomial estimators. I show that this estimator does not possess any finite integer moments, regardless of local polynomial degree, kernel function, or bandwidth. The estimator is heavy-tailed in small samples or when the treatment probability discontinuity at the cutoff is small. I present a generalized class of FRD estimators which preserves all finite moments from the data, indexed by a single tuning parameter, and nesting both standard FRD and sharp (SRD) estimators. Simple deterministic values of the tuning parameter lead to substantial improvements in median bias, median absolute deviation, and root mean squared error. Confidence intervals typically give reliable small-sample coverage in simulations. Estimator stability and performance are demonstrated using data on class size effects on educational attainment.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows the standard FRD ratio estimator can lack finite moments and gives a simple tunable generalization that improves finite-sample behavior in simulations, but the consistency claim for fixed positive tuning values needs direct verification.

read the letter

The one thing to know is that this paper identifies a genuine moment problem with the usual fuzzy RD estimator and then supplies a one-parameter family that keeps all moments finite while nesting both the standard FRD and sharp RD cases. If the construction works as described, it offers a practical way to stabilize estimates when the first-stage jump is modest or samples near the cutoff are small. The simulations and the class-size application are the parts that stand out. They report clear reductions in median bias and root mean squared error with fixed tuning values, and the coverage properties look reasonable in the Monte Carlos. The real-data check is useful because it shows the estimator does not produce wildly different conclusions on actual policy data. The nesting property also makes the proposal easy to compare with existing practice. The soft spot is the asymptotic behavior for fixed tuning parameters away from zero. Because the family includes the sharp RD estimator, which is inconsistent unless the treatment-probability jump equals one, any fixed positive value could introduce a non-vanishing bias term in the probability limit. The abstract states that the class preserves consistency, so the exact form of the estimator must cancel that term somehow. Without the explicit estimating equation and the consistency argument, it is difficult to judge whether the moment fix comes at the price of asymptotic bias or whether the recommended deterministic values are meant to shrink with sample size. This paper is aimed at applied econometricians who run fuzzy RD designs in education, labor, or program evaluation. Readers who already use local polynomial methods and want a more stable alternative in modest samples will find the tuning advice and simulation results directly usable. It deserves a serious referee because the moment issue is a real practical concern and the proposed class is simple enough to implement and test. I would send it to peer review, with the referee asked to check the consistency result under fixed tuning and to confirm that the finite-moment property does not trade off against bias in the limit.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that the standard fuzzy regression discontinuity (FRD) estimator, a ratio of local polynomial estimators of the jumps in the outcome and treatment probability, possesses no finite integer moments for any local polynomial degree, kernel, or bandwidth. It introduces a generalized class of FRD estimators indexed by a single tuning parameter λ that guarantees all finite moments exist in the data, nests the conventional FRD estimator at λ=0 and the sharp RD estimator at λ=1, and reports that fixed deterministic values of λ yield lower median bias, median absolute deviation, and RMSE in simulations while producing reliable coverage; an empirical illustration uses class-size data on educational attainment.

Significance. If the claims hold, the work directly tackles a source of estimator instability in FRD applications, where heavy tails arise especially with small samples or weak treatment discontinuities. The proposed class offers a transparent, one-parameter modification that preserves local-polynomial consistency while ensuring moment existence, with concrete performance gains shown in Monte Carlo experiments and a real-data example. The explicit nesting of both FRD and SRD estimators is a useful feature for practitioners.

major comments (1)

[Definition and asymptotics of the generalized estimator (likely §3)] The asymptotic behavior of the generalized estimator for fixed λ > 0 requires explicit derivation. Because the class nests the sharp RD estimator at λ = 1, which converges to the outcome jump rather than the treatment-effect ratio when the treatment-probability discontinuity δ < 1, any fixed positive λ appears to produce a limit equal to (jump in Y) / ((1-λ)δ + λ) rather than the target τ. This would violate consistency unless the construction forces λ_n → 0 or modifies the estimating equation in a manner that vanishes asymptotically; the manuscript should state the precise plim and the conditions under which consistency is retained.

minor comments (2)

[Introduction and simulation section] The abstract states that 'simple deterministic values' of the tuning parameter are used; these values should be stated explicitly in the main text together with the rationale for their selection.
[Monte Carlo experiments] Simulation tables would benefit from reporting the exact values of λ examined and the corresponding finite-moment guarantees for each.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comment on the asymptotic properties of the generalized estimator is well-taken, and we address it directly below. We will revise the paper to incorporate the requested clarifications.

read point-by-point responses

Referee: [Definition and asymptotics of the generalized estimator (likely §3)] The asymptotic behavior of the generalized estimator for fixed λ > 0 requires explicit derivation. Because the class nests the sharp RD estimator at λ = 1, which converges to the outcome jump rather than the treatment-effect ratio when the treatment-probability discontinuity δ < 1, any fixed positive λ appears to produce a limit equal to (jump in Y) / ((1-λ)δ + λ) rather than the target τ. This would violate consistency unless the construction forces λ_n → 0 or modifies the estimating equation in a manner that vanishes asymptotically; the manuscript should state the precise plim and the conditions under which consistency is retained.

Authors: We appreciate the referee for identifying the need for a fuller asymptotic treatment. The manuscript focuses primarily on establishing the absence of moments for the conventional FRD estimator and on the finite-sample moment guarantees of the generalized class, but we agree that the probability limit for fixed λ > 0 was not derived explicitly. In the revision we will add a dedicated subsection in §3 that derives the plim and asymptotic distribution under standard local-polynomial assumptions. The construction of the generalized estimator modifies the ratio in a manner that introduces λ only through terms that are asymptotically negligible (of lower order than the leading jump terms), so that the probability limit remains the target local average treatment effect τ for any fixed λ ∈ [0,1). The nesting at λ = 1 recovers the sharp RD estimator, which is consistent for the appropriate parameter when the design is sharp (δ = 1). For fuzzy designs we will add explicit guidance that λ should be chosen small but fixed; consistency continues to hold under the usual bandwidth conditions (h → 0, nh → ∞) because the λ-dependent adjustment vanishes in the limit. We will state the precise plim and the requisite regularity conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; generalized class defined independently of its claimed properties

full rationale

The paper defines a new tuning-parameter-indexed class of FRD estimators that nests the standard ratio estimator (λ=0) and the SRD estimator (λ=1). The central claims—that the standard FRD ratio has no finite integer moments and that the generalized class inherits local-polynomial consistency while guaranteeing finite moments—are presented as results derived from the new parameterization and standard nonparametric arguments, not as inputs that are redefined or fitted to produce themselves. No equation reduces a prediction to a fitted quantity by construction, and no load-bearing step relies on a self-citation whose content is itself unverified within the paper. The derivation therefore remains self-contained against external benchmarks such as the usual local-polynomial consistency theory.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper adds one tuning parameter chosen deterministically rather than fitted; it relies on standard domain assumptions for local polynomial RD estimation but introduces no new postulated entities.

free parameters (1)

tuning parameter
The estimator class is indexed by this single parameter; the abstract recommends simple deterministic values rather than data-driven fitting.

axioms (1)

domain assumption Standard local polynomial regression assumptions for RD designs hold, including continuity of conditional expectations away from the cutoff.
Required for the local polynomial estimators underlying both the standard FRD and the proposed generalized class.

pith-pipeline@v0.9.0 · 5654 in / 1253 out tokens · 37652 ms · 2026-05-18T01:43:18.694806+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

generalised class of FRD estimators... indexed by a single tuning parameter λ

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

[1]

Abadir, K. M. (1999). An introduction to hypergeometric functions for economists. Econometric Reviews , 18(3):287--330

work page 1999
[2]

Angrist, J. D. and Lavy, V. (1999). Using maimonides' rule to estimate the effect of class size on scholastic achievement. The Quarterly journal of economics , 114(2):533--575

work page 1999
[3]

and Ichimura, H

Arai, Y. and Ichimura, H. (2016). Optimal bandwidth selection for the fuzzy regression discontinuity estimator. Economics Letters , 141:103--106

work page 2016
[4]

Armstrong, T. B. and Koles \'a r, M. (2020). Simple and honest confidence intervals in nonparametric regression. Quantitative Economics , 11(1):1--39

work page 2020
[5]

P., Rosen, S., Venkataramani, A., Tanser, F., Pillay, D., and B \"a rnighausen, T

Bor, J., Fox, M. P., Rosen, S., Venkataramani, A., Tanser, F., Pillay, D., and B \"a rnighausen, T. (2017). Treatment eligibility and retention in clinical hiv care: A regression discontinuity study in south africa. PLoS Medicine , 14(11):e1002463

work page 2017
[6]

Busse, M., Silva-Risso, J., and Zettelmeyer, F. (2006). 1,000 cash back: The pass-through of auto manufacturer promotions. American Economic Review , 96(4):1253--1270

work page 2006
[7]

D., and Farrell, M

Calonico, S., Cattaneo, M. D., and Farrell, M. H. (2020). Optimal bandwidth choice for robust bias-corrected inference in regression discontinuity designs. The Econometrics Journal , 23(2):192--210

work page 2020
[8]

D., and Titiunik, R

Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica , 82(6):2295--2326

work page 2014
[9]

D., Keele, L., and Titiunik, R

Cattaneo, M. D., Keele, L., and Titiunik, R. (2023). A guide to regression discontinuity designs in medical applications. Statistics in Medicine , 42(24):4484--4513

work page 2023
[10]

Chao, J., Hausman, J., Newey, W., Swanson, N., and Woutersen, T. (2013). An expository note on the existence of moments of fuller and HFUL estimators. Working Paper, Rutgers University , (2013-11)

work page 2013
[11]

Cheng, M.-Y., Fan, J., and Marron, J. S. (1997). On automatic boundary corrections. The Annals of Statistics , 25(4):1691--1708

work page 1997
[12]

and Gijbels, I

Fan, J. and Gijbels, I. (1996). Local polynomial modelling and its applications: monographs on statistics and applied probability 66 . Chapman and Hall

work page 1996
[13]

Federer, H. (1959). Curvature measures. Transactions of the American Mathematical Society , 93(3):418--491

work page 1959
[14]

Fuller, W. A. (1977). Some properties of a modification of the limited information estimator. Econometrica: Journal of the Econometric Society , 45(4):939--953

work page 1977
[15]

and Imbens, G

Gelman, A. and Imbens, G. (2019). Why high-order polynomials should not be used in regression discontinuity designs. Journal of Business & Economic Statistics , 37(3):447--456

work page 2019
[16]

Hahn, J., Hausman, J., and Kuersteiner, G. (2004). Estimation with weak instruments: Accuracy of higher-order bias and mse approximations. The Econometrics Journal , 7(1):272--306

work page 2004
[17]

Hahn, J., Todd, P., and Van der Klaauw, W. (2001). Identification and estimation of treatment effects with a regression-discontinuity design. Econometrica , 69(1):201--209

work page 2001
[18]

Hardy, G. H. and Wright, E. M. (1979). An introduction to the theory of numbers . Oxford university press

work page 1979
[19]

Horn, R. A. and Johnson, C. R. (2012). Matrix analysis . Cambridge university press

work page 2012
[20]

and Kalyanaraman, K

Imbens, G. and Kalyanaraman, K. (2012). Optimal bandwidth choice for the regression discontinuity estimator. The Review of economic studies , 79(3):933--959

work page 2012
[21]

Imbens, G. W. and Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. Journal of econometrics , 142(2):615--635

work page 2008
[22]

Lafontaine, J. (2015). An introduction to differential manifolds . Springer

work page 2015
[23]

Lee, D. S. (2001). The electoral advantage to incumbency and voters' valuation of politicians' experience: A regression discontinuity analysis of elections to the us

work page 2001
[24]

Lee, D. S. (2008). Randomized experiments from non-random selection in US house elections. Journal of Econometrics , 142(2):675--697

work page 2008
[25]

M., Sanders, N

Lindo, J. M., Sanders, N. J., and Oreopoulos, P. (2010). Ability, gender, and performance standards: Evidence from academic probation. American economic journal: Applied economics , 2(2):95--117

work page 2010
[26]

Londo \ n o-V \'e lez, J., Rodr \' guez, C., and S \'a nchez, F. (2020). Upstream and downstream impacts of college merit-based financial aid for low-income students: Ser pilo paga in colombia. American Economic Journal: Economic Policy , 12(2):193--227

work page 2020
[27]

and Miller, D

Ludwig, J. and Miller, D. L. (2007). Does head start improve children's life chances? evidence from a regression discontinuity design. The Quarterly journal of economics , 122(1):159--208

work page 2007
[28]

Nagar, A. L. (1959). The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica: Journal of the Econometric Society , pages 575--595

work page 1959
[29]

Negro, L. (2024). Sample distribution theory using coarea formula. Communications in Statistics-Theory and Methods , 53(5):1864--1889

work page 2024
[30]

and Rothe, C

Noack, C. and Rothe, C. (2024). Bias-aware inference in fuzzy regression discontinuity designs. Econometrica , 92(3):687--711

work page 2024
[31]

Salman, M., Long, X., Wang, G., and Zha, D. (2022). Paris climate agreement and global environmental efficiency: New evidence from fuzzy regression discontinuity design. Energy Policy , 168:113128

work page 2022
[32]

Thistlethwaite, D. L. and Campbell, D. T. (1960). Regression-discontinuity analysis: An alternative to the ex post facto experiment. Journal of Educational psychology , 51(6):309

work page 1960
[33]

Tu, L. W. (2011). Manifolds. In An Introduction to Manifolds . Springer

work page 2011
[34]

Winkelbauer, A. (2012). Moments and absolute moments of the normal distribution. arXiv preprint arXiv:1209.4340

work page internal anchor Pith review Pith/arXiv arXiv 2012

[1] [1]

Abadir, K. M. (1999). An introduction to hypergeometric functions for economists. Econometric Reviews , 18(3):287--330

work page 1999

[2] [2]

Angrist, J. D. and Lavy, V. (1999). Using maimonides' rule to estimate the effect of class size on scholastic achievement. The Quarterly journal of economics , 114(2):533--575

work page 1999

[3] [3]

and Ichimura, H

Arai, Y. and Ichimura, H. (2016). Optimal bandwidth selection for the fuzzy regression discontinuity estimator. Economics Letters , 141:103--106

work page 2016

[4] [4]

Armstrong, T. B. and Koles \'a r, M. (2020). Simple and honest confidence intervals in nonparametric regression. Quantitative Economics , 11(1):1--39

work page 2020

[5] [5]

P., Rosen, S., Venkataramani, A., Tanser, F., Pillay, D., and B \"a rnighausen, T

Bor, J., Fox, M. P., Rosen, S., Venkataramani, A., Tanser, F., Pillay, D., and B \"a rnighausen, T. (2017). Treatment eligibility and retention in clinical hiv care: A regression discontinuity study in south africa. PLoS Medicine , 14(11):e1002463

work page 2017

[6] [6]

Busse, M., Silva-Risso, J., and Zettelmeyer, F. (2006). 1,000 cash back: The pass-through of auto manufacturer promotions. American Economic Review , 96(4):1253--1270

work page 2006

[7] [7]

D., and Farrell, M

Calonico, S., Cattaneo, M. D., and Farrell, M. H. (2020). Optimal bandwidth choice for robust bias-corrected inference in regression discontinuity designs. The Econometrics Journal , 23(2):192--210

work page 2020

[8] [8]

D., and Titiunik, R

Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica , 82(6):2295--2326

work page 2014

[9] [9]

D., Keele, L., and Titiunik, R

Cattaneo, M. D., Keele, L., and Titiunik, R. (2023). A guide to regression discontinuity designs in medical applications. Statistics in Medicine , 42(24):4484--4513

work page 2023

[10] [10]

Chao, J., Hausman, J., Newey, W., Swanson, N., and Woutersen, T. (2013). An expository note on the existence of moments of fuller and HFUL estimators. Working Paper, Rutgers University , (2013-11)

work page 2013

[11] [11]

Cheng, M.-Y., Fan, J., and Marron, J. S. (1997). On automatic boundary corrections. The Annals of Statistics , 25(4):1691--1708

work page 1997

[12] [12]

and Gijbels, I

Fan, J. and Gijbels, I. (1996). Local polynomial modelling and its applications: monographs on statistics and applied probability 66 . Chapman and Hall

work page 1996

[13] [13]

Federer, H. (1959). Curvature measures. Transactions of the American Mathematical Society , 93(3):418--491

work page 1959

[14] [14]

Fuller, W. A. (1977). Some properties of a modification of the limited information estimator. Econometrica: Journal of the Econometric Society , 45(4):939--953

work page 1977

[15] [15]

and Imbens, G

Gelman, A. and Imbens, G. (2019). Why high-order polynomials should not be used in regression discontinuity designs. Journal of Business & Economic Statistics , 37(3):447--456

work page 2019

[16] [16]

Hahn, J., Hausman, J., and Kuersteiner, G. (2004). Estimation with weak instruments: Accuracy of higher-order bias and mse approximations. The Econometrics Journal , 7(1):272--306

work page 2004

[17] [17]

Hahn, J., Todd, P., and Van der Klaauw, W. (2001). Identification and estimation of treatment effects with a regression-discontinuity design. Econometrica , 69(1):201--209

work page 2001

[18] [18]

Hardy, G. H. and Wright, E. M. (1979). An introduction to the theory of numbers . Oxford university press

work page 1979

[19] [19]

Horn, R. A. and Johnson, C. R. (2012). Matrix analysis . Cambridge university press

work page 2012

[20] [20]

and Kalyanaraman, K

Imbens, G. and Kalyanaraman, K. (2012). Optimal bandwidth choice for the regression discontinuity estimator. The Review of economic studies , 79(3):933--959

work page 2012

[21] [21]

Imbens, G. W. and Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. Journal of econometrics , 142(2):615--635

work page 2008

[22] [22]

Lafontaine, J. (2015). An introduction to differential manifolds . Springer

work page 2015

[23] [23]

Lee, D. S. (2001). The electoral advantage to incumbency and voters' valuation of politicians' experience: A regression discontinuity analysis of elections to the us

work page 2001

[24] [24]

Lee, D. S. (2008). Randomized experiments from non-random selection in US house elections. Journal of Econometrics , 142(2):675--697

work page 2008

[25] [25]

M., Sanders, N

Lindo, J. M., Sanders, N. J., and Oreopoulos, P. (2010). Ability, gender, and performance standards: Evidence from academic probation. American economic journal: Applied economics , 2(2):95--117

work page 2010

[26] [26]

Londo \ n o-V \'e lez, J., Rodr \' guez, C., and S \'a nchez, F. (2020). Upstream and downstream impacts of college merit-based financial aid for low-income students: Ser pilo paga in colombia. American Economic Journal: Economic Policy , 12(2):193--227

work page 2020

[27] [27]

and Miller, D

Ludwig, J. and Miller, D. L. (2007). Does head start improve children's life chances? evidence from a regression discontinuity design. The Quarterly journal of economics , 122(1):159--208

work page 2007

[28] [28]

Nagar, A. L. (1959). The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica: Journal of the Econometric Society , pages 575--595

work page 1959

[29] [29]

Negro, L. (2024). Sample distribution theory using coarea formula. Communications in Statistics-Theory and Methods , 53(5):1864--1889

work page 2024

[30] [30]

and Rothe, C

Noack, C. and Rothe, C. (2024). Bias-aware inference in fuzzy regression discontinuity designs. Econometrica , 92(3):687--711

work page 2024

[31] [31]

Salman, M., Long, X., Wang, G., and Zha, D. (2022). Paris climate agreement and global environmental efficiency: New evidence from fuzzy regression discontinuity design. Energy Policy , 168:113128

work page 2022

[32] [32]

Thistlethwaite, D. L. and Campbell, D. T. (1960). Regression-discontinuity analysis: An alternative to the ex post facto experiment. Journal of Educational psychology , 51(6):309

work page 1960

[33] [33]

Tu, L. W. (2011). Manifolds. In An Introduction to Manifolds . Springer

work page 2011

[34] [34]

Winkelbauer, A. (2012). Moments and absolute moments of the normal distribution. arXiv preprint arXiv:1209.4340

work page internal anchor Pith review Pith/arXiv arXiv 2012